scholarly journals Kurosh-Amitsur Right Jacobson Radical of Type 0 for Right Near-Rings

2008 ◽  
Vol 2008 ◽  
pp. 1-6
Author(s):  
Ravi Srinivasa Rao ◽  
K. Siva Prasad ◽  
T. Srinivas
Keyword(s):  
Jacobson Radical ◽  
Paper Properties ◽  
Hereditary Radical ◽  
Constant Part ◽  
The Right

By a near-ring we mean a right near-ring.J0r, the right Jacobson radical of type 0, was introduced for near-rings by the first and second authors. In this paper properties of the radicalJ0rare studied. It is shown thatJ0ris a Kurosh-Amitsur radical (KA-radical) in the variety of all near-ringsR, in which the constant partRcofRis an ideal ofR. So unlike the left Jacobson radicals of types 0 and 1 of near-rings,J0ris a KA-radical in the class of all zero-symmetric near-rings.J0ris nots-hereditary and hence not an ideal-hereditary radical in the class of all zero-symmetric near-rings.

scholarly journals On p-injective rings

1995 ◽  
Vol 37 (3) ◽  
pp. 373-378 ◽  
Author(s):  
Gennadi Puninski ◽  
Robert Wisbauer ◽  
Mohamed Yousif
Keyword(s):  
Direct Summand ◽  
Jacobson Radical ◽  
Associative Ring ◽  
The Right ◽  
Left Annihilator

Throughout this paper R will be an associative ring with unity and all R-modules are unitary. The right (resp. left) annihilator in R of a subset X of a module is denoted by r(X)(resp. I(X)). The Jacobson radical of R is denoted by J(R), the singular ideals are denoted by Z(RR) and Z(RR) and the socles by Soc(RR) and Soc(RR). For a module M, E(M) and PE(M) denote the injective and pure-injective envelopes of M, respectively. For a submodule A ⊆ M, the notation A ⊆⊕M will mean that A is a direct summand of M.

ON SMALL INJECTIVE RINGS AND MODULES

2009 ◽  
Vol 08 (03) ◽  
pp. 379-387 ◽  
Author(s):  
LE VAN THUYET ◽  
TRUONG CONG QUYNH
Keyword(s):  
Jacobson Radical ◽  
Injective Module ◽  
The Right

A right R-module MR is called small injective if every homomorphism from a small right ideal to MR can be extended to an R-homomorphism from RR to MR. A ring R is called right small injective, if the right R-module RR is small injective. We prove that R is semiprimitive if and only if every simple right (or left) R-module is small injective. Further we show that the Jacobson radical J of a ring R is a noetherian right R-module if and only if, for every small injective module ER, E(ℕ) is small injective.

scholarly journals Coincidence of topological Jacobson radicals in topological algebras

2017 ◽  
Vol 21 (2) ◽  
pp. 239-247
Author(s):  
Mart Abel ◽  
Mati Abel ◽  
Paul Tammo
Keyword(s):  
Jacobson Radical ◽  
Topological Algebras ◽  
The Right

Several classes of topological algebras for which the left topological Jacobson radical coincides with the right topological Jacobson radical are described.

scholarly journals ON 𝒯-NONCOSINGULAR MODULES

2009 ◽  
Vol 80 (3) ◽  
pp. 462-471 ◽  
Author(s):  
DERYA KESKIN TÜTÜNCÜ ◽  
RACHID TRIBAK
Keyword(s):  
Jacobson Radical ◽  
The Right

AbstractIn this paper we introduce 𝒯-noncosingular modules. Rings for which all right modules are𝒯-noncosingular are shown to be precisely those for which every simple right module is injective. Moreover, for any ring R we show that the right R-module R is 𝒯-noncosingular precisely when R has zero Jacobson radical. We also study the 𝒯-noncosingular condition in association with (strongly) FI-lifting modules.

scholarly journals On the radical of a group algebra over a commutative ring

1977 ◽  
Vol 18 (1) ◽  
pp. 101-104 ◽  
Author(s):  
Gloria Potter
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Group Algebra ◽  
Jacobson Radical ◽  
Multiplicative Group ◽  
Commutative Rings ◽  
The Other ◽  
Group Algebras ◽  
Study Group ◽  
The Right

Several people, including Wallace [4] and Passman [3], have studied the Jacobson radical of the group algebra F[G] where F is a field and G is a multiplicative group. In [4], for instance, Wallace proves that if G is an abelian group with Sylow p-subgroup P and if F is a field of characteristic p, then the Jacobson radical of F[G] equals the right ideal generated by the radical of F[P]. In this paper we shall study group algebras over arbitrary commutative rings. By a reduction to the case of a semi-simple commutative ring, we obtain Theorem 1 whose corollary contains a generalization of Wallace's theorem. Theorem 2, on the other hand, uses the first theorem to obtain results related to the main theorem of [3].

scholarly journals A radical for right near-rings: The right Jacobson radical of type-0

2006 ◽  
Vol 2006 ◽  
pp. 1-13 ◽  
Author(s):  
Ravi Srinivasa Rao ◽  
K. Siva Prasad
Keyword(s):  
Jacobson Radical ◽  
The Right ◽  
Nil Radical

The notions of a right quasiregular element and right modular right ideal in a near-ring are initiated. Based on theseJ0r(R), the right Jacobson radical of type-0 of a near-ringRis introduced. It is obtained thatJ0ris a radical map andN(R)⊆J0r(R), whereN(R)is the nil radical of a near-ringR. Some characterizations ofJ0r(R) are given and its relation with some of the radicals is also discussed.

Generalized Discrete Valuation Rings

1969 ◽  
Vol 21 ◽  
pp. 1404-1408 ◽  
Author(s):  
H.-H. Brungs
Keyword(s):  
Noetherian Ring ◽  
Jacobson Radical ◽  
Unit Element ◽  
Global Dimension ◽  
Order Type ◽  
Unique Factorization ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
The Right ◽  
Satisfy Property

Jategaonkar (5) has constructed a class of rings which can be used to provide counterexamples to problems concerning unique factorization in non-commutative domains, the left-right symmetry of the global dimension for a right- Noetherian ring and the transhnite powers of the Jacobson radical of a right- Noetherian ring. These rings have the following property:(W) Every non-empty family of right ideals of the ring R contains exactly one maximal element.In the present paper we wish to consider rings, with unit element, which satisfy property (W). This property means that the right ideals are inverse well-ordered by inclusion, and it is our aim to describe these rings by their order type. Rings of this kind appear as a generalization of discrete valuation rings in R; see (1; 2).In the following, R will always denote a ring with unit element satisfying (W).

On the Yoneda Ext-Algebras of Semiperfect Algebras

2008 ◽  
Vol 15 (02) ◽  
pp. 207-222 ◽  
Author(s):  
Jiwei He ◽  
Yu Ye
Keyword(s):  
Jacobson Radical ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Yoneda Algebra ◽  
Ideal Formula ◽  
Morita Equivalent ◽  
Morita Invariant ◽  
The Right ◽  
Necessary And Sufficient

It is proved that the Yoneda Ext-algebras of Morita equivalent semiperfect algebras are graded equivalent. The Yoneda Ext-algebras of noetherian semiperfect algebras are studied in detail. Let A be a noetherian semiperfect algebra with Jacobson radical J. We construct a right ideal [Formula: see text] of the Yoneda algebra [Formula: see text], which plays an important role in the discussion of the structure of E(A). An extra grading is introduced to [Formula: see text], by which we give a description of the right ideal of E(A) generated by [Formula: see text], and we give a necessary and sufficient condition for a notherian semiperfect algebra to be higher quasi-Koszul. Finally, it is shown that the quasi-Koszulity of a noetherian semiperfect algebra is a Morita invariant.

A Generalization of Semiregular and Almost Principally Injective Rings

2010 ◽  
Vol 17 (spec01) ◽  
pp. 905-916
Author(s):  
A. Çiğdem Özcan ◽  
Pınar Aydoğdu
Keyword(s):  
Jacobson Radical ◽  
Left And Right ◽  
Singular Ideal ◽  
The Right ◽  
The Ideal

In this article, we call a ring R right almost I-semiregular for an ideal I of R if for any a ∈ R, there exists a left R-module decomposition lRrR(a) = P ⊕ Q such that P ⊆ Ra and Q ∩ Ra ⊆ I, where l and r are the left and right annihilators, respectively. This generalizes the right almost principally injective rings defined by Page and Zhou, I-semiregular rings defined by Nicholson and Yousif, and right generalized semiregular rings defined by Xiao and Tong. We prove that R is I-semiregular if and only if for any a ∈ R, there exists a decomposition lRrR(a) = P ⊕ Q, where P = Re ⊆ Ra for some e2 = e ∈ R and Q ∩ Ra ⊆ I. Among the results for right almost I-semiregular rings, we show that if I is the left socle Soc (RR) or the right singular ideal Z(RR) or the ideal Z(RR) ∩ δ(RR), where δ(RR) is the intersection of essential maximal left ideals of R, then R being right almost I-semiregular implies that R is right almost J-semiregular for the Jacobson radical J of R. We show that δl(eRe) = e δ(RR)e for any idempotent e of R satisfying ReR = R and, for such an idempotent, R being right almost δ(RR)-semiregular implies that eRe is right almost δl(eRe)-semiregular.

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